Question: A composite function can be written as $w\bigl(u(x)\bigr)$, where $u$ and $w$ are basic functions. Is $f(x)=\ln(\sin(x))$ a composite function? If so, what are $u$ and $w$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $f$ is composite. $u(x)=\sin(x)$ and $w(x)=\ln(x)$. (Choice B) B $f$ is composite. $u(x)=\ln(x)$ and $w(x)=\sin(x)$. (Choice C) C $f$ is not a composite function.
Solution: Composite and combined functions A composite function is where we make the output from one function, in this case $u$, the input for another function, in this case $w$. We can also combine functions using arithmetic operations, but such a combination is not considered a composite function. The inner function The inner function is the part we evaluate first. Frequently, we can identify the correct expression because it will appear within a grouping symbol one or more times in our composed function. Here, we have two sets of parentheses. The innermost only has $x$ inside of it, so we look to the next grouping symbol. We find $\sin(x)$ inside the next pair of parentheses. We evaluate this expression first, so $u(x)=\sin(x)$ is the inner function. The outer function Then we take the natural logarithm of the entire output of $u$. So $w(x)=\ln(x)$ is the outer function. Answer $f$ is composite. $u(x)=\sin(x)$ and $w(x)=\ln(x)$. Note that there are other valid ways to decompose $f$, especially into more complicated functions.